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Percents and Applications

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Percents and Applications - Complete Interactive Lesson

Part 1: What a Percent Really Means

💯 Percents and Applications

Part 1 of 5 — What a Percent Really Means

Topics in This Part

Section
Percent = "Per Hundred"
Converting Percents ↔ Decimals
Converting Percents ↔ Fractions

🔑 Key Concept: A percent is just a fraction with a denominator of $100$ . The word percent comes from the Latin per centum — "per hundred." Master this one idea and every percent problem becomes a fraction problem.

Percent = "Per Hundred"

When you see $45\%$ , read it as 45 out of 100:

$45\% = \frac{45}{100} = 0.45$

Picture a $10 \times 10$ grid of $100$ squares. Shading $45$ of them shows $45\%$ — almost half the grid.

Percent	Out of 100	As a fraction
$25\%$	$25$ shaded	$\dfrac{25}{100} = \dfrac{1}{4}$

💡 A percent can be more than $100$ . $100\%$ means "the whole thing," so $150\%$ means "one and a half times as much."

Concept Check 🎯

Percents ↔ Decimals

Because a percent is "out of $100$ ," converting is all about moving the decimal point two places.

Percent → Decimal: drop the $\%$ sign and move the decimal 2 places left (÷ 100).

$45\% \to 0.45 \qquad 7\% \to 0.07 \qquad 120\% \to 1.20$

Convert Percents and Decimals 🧮

Move the decimal two places.

1) $45\%$ as a decimal $= \,?$ 2) $7\%$ as a decimal $= \,?$ 3) $0.6$ as a percent

Percents ↔ Fractions

Percent → Fraction: put the number over $100$ , then simplify.

$75\% = \frac{75}{100} = \frac{3}{4} \qquad 40\% = \frac{40}{100} = \frac{2}{5}$

Match the Forms 🔽

Choose the correct equivalent for each.

Part 2: Finding a Percent of a Number

💯 Percents and Applications

Part 2 of 5 — Finding a Percent of a Number

🔑 The Core Skill: " $20\%$ of $80$ " is a multiplication problem. The word "of" means multiply, and a percent is a decimal in disguise:

$20\% \text{ of } 80 = 0.20 \times 80 = 16$

Part 3: Percent Increase & Decrease

💯 Percents and Applications

Part 3 of 5 — Percent Increase & Decrease

🔑 The Idea: When a quantity changes, the percent change tells you how big the change is relative to where it started:

$\text{percent change} = \frac{\text{amount of change}}{\text{original amount}} \times 100\%$

Part 4: Real-World Applications

💯 Percents and Applications

Part 4 of 5 — Real-World Applications

🔑 Same skill, new names. Sales tax, tips, commission, and simple interest are all "percent of a number" problems wearing different costumes.

Sales Tax and Tips

Both tax and tip are amounts added on top of a base price.

$\text{tax (or tip)} = \text{rate} \times \text{base price}$ $\text{total} = \text{base} + \text{tax (or tip)}$

Part 5: Mixed Practice & Mastery Check

💯 Percents and Applications

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) convert between percents, decimals, and fractions, (2) find a percent of a number and solve all three percent questions, (3) handle percent increase and decrease, and (4) apply percents to tax, tip, commission, and interest. Let's put it all together.

Quick Reference

Goal	Key move
Percent → decimal	move decimal $2$ places left ( $45\% \to 0.45$ )
Decimal → percent	move decimal $2$ places ()

Decimal → Percent: move the decimal 2 places right and add a $\%$ sign (× 100).

$0.6 \to 60\% \qquad 0.03 \to 3\% \qquad 2.5 \to 250\%$

⚠️ Watch single digits. $7\%$ is $0.07$ , not $0.7$ . You must fill an empty place with a zero: $07. \to 0.07$ .

(enter just the number)

Fraction → Percent: divide, then move the decimal $2$ places right (or build an equivalent fraction with denominator $100$ ).

$\frac{1}{8} = 0.125 = 12.5\% \qquad \frac{3}{4} = \frac{75}{100} = 75\%$

Memorize These Benchmarks

Fraction	Decimal	Percent
$\dfrac{1}{2}$	$0.5$	$50\%$
$\dfrac{1}{4}$	$0.25$	$25\%$
$\dfrac{3}{4}$	$0.75$	$75\%$
$\dfrac{1}{5}$	$0.2$	$20\%$
$\dfrac{1}{8}$	$0.125$	$12.5\%$

🔑 Knowing these by heart makes mental percents fast: $25\%$ of anything is just one quarter of it.

"Of" Means Multiply

To find a percent of a number, change the percent to a decimal and multiply:

$\text{(percent as decimal)} \times \text{(the number)} = \text{the part}$

Worked Example: $15\%$ of $60$

Convert: $15\% = 0.15$
Multiply: $0.15 \times 60 = 9$

So $15\%$ of $60$ is $\mathbf{9}$ .

Worked Example: $8\%$ of $250$

$8\% = 0.08 \quad\Rightarrow\quad 0.08 \times 250 = 20$

Worked Example: $150\%$ of $40$

$150\% = 1.50 \quad\Rightarrow\quad 1.50 \times 40 = 60$

💡 Because $150\% > 100\%$ , the answer ( $60$ ) is bigger than the original number ( $40$ ). That's a good sanity check.

Concept Check 🎯

Estimate to Catch Mistakes

Before you trust a calculation, estimate with a benchmark percent:

$10\%$ of a number is just that number with the decimal moved one place left: $10\%$ of $60$ is $6$ .
So $15\%$ of $60$ should be a bit more than $6$ — and indeed it's $9$ . ✓

💡 If your answer is wildly off your estimate, you probably forgot to convert the percent to a decimal. ( $15 \times 60 = 900$ is a giant red flag.)

Find the Part 🧮

Convert to a decimal and multiply.

1) $15\%$ of $60 = \,?$ 2) $8\%$ of $250 = \,?$ 3) $150\%$ of $40 = \,?$

The Three Percent Questions

Every percent problem is one of three types. They all come from the same relationship:

$\text{part} = \text{percent} \times \text{whole}$

Question type	What's missing	How to solve
Find the part	the part	multiply: $0.25 \times 80 = 20$
Find the percent	the percent	divide part by whole: $\dfrac{18}{24} = 0.75 = 75\%$
Find the whole	the whole	divide part by percent: $\dfrac{30}{0.25} = 120$

Worked Example: "What percent of $24$ is $18$ ?"

$\frac{\text{part}}{\text{whole}} = \frac{18}{24} = 0.75 = 75\%$

Worked Example: " $30$ is $25\%$ of what number?"

$\text{whole} = \frac{\text{part}}{\text{percent}} = \frac{30}{0.25} = 120$

⚠️ Read carefully. "What percent" → divide part by whole. "Of what number" → divide part by the decimal percent. Mixing these up is the #1 percent mistake.

Pick the Move 🔽

For each question, choose the calculation that solves it.

Decoding the Words

The phrasing tells you which operation to use:

The sentence says...	You are finding the...	So you...
"What is X% of N"	part	multiply
"What percent of N is P"	percent	divide $\dfrac{P}{N}$
"P is X% of what"	whole	divide $\dfrac{P}{\text{decimal}}$

🔑 Translate the words first, then compute. Now solve one of each type yourself.

Solve Each Type 🧮

1) What percent of $24$ is $18$ ? $\,?\%$ (enter just the number) 2) $30$ is $25\%$ of what number? $\,?$ 3) What is $5\%$ of $200$ ? $\,?$

Finding the Percent Change

Two steps:

Amount of change $=$ new value $-$ original value.
Divide by the original, then convert to a percent.

Worked Example: A price rises from $40$ to $50$

$\text{change} = 50 - 40 = 10$ $\frac{10}{40} = 0.25 = 25\% \text{ increase}$

Worked Example: A price drops from $80$ to $60$

$\text{change} = 60 - 80 = -20$ $\frac{-20}{80} = -0.25 = 25\% \text{ decrease}$

⚠️ Always divide by the original value, not the new one. Going $40 \to 50$ is a $25\%$ increase, but $50 \to 40$ is a $20\%$ decrease — different originals, different percents.

Concept Check 🎯

Markups and Discounts

Stores use percent change constantly: a markup raises a price, a discount lowers it.

There are two equally good methods.

Method A — Find the change, then add or subtract

A shirt costs $25 and is marked up $20\%$ :

$\text{markup} = 0.20 \times 25 = 5 \quad\Rightarrow\quad 25 + 5 = \$30$

A $80 jacket is $30\%$ off:

$\text{discount} = 0.30 \times 80 = 24 \quad\Rightarrow\quad 80 - 24 = \$56$

Method B — Multiply by the final percent (the shortcut)

A $20\%$ increase means you keep $100\% + 20\% = 120\%$ : $\;1.20 \times 25 = \$ 30$.
A $30\%$ discount means you pay $100\% - 30\% = 70\%$ : 56$.

💡 Method B is faster. For a $30\%$ -off sale, just multiply by $0.70$ — you pay $70\%$ of the original. One step, same answer.

The One-Step Multiplier 🔽

Choose the single number you'd multiply the original price by.

One Multiplier, Final Answer

You just built every shortcut multiplier. Putting Method B to work:

$\underbrace{1.20}_{+20\%} \times 25 = \$30 \qquad \underbrace{0.70}_{-30\%} \times 80 = \$56$

💡 Method B (multiply once) and Method A (find the change, then add/subtract) always give the same answer. Use whichever you find clearer — but Method B is fewer steps. Try a few on your own.

Markups and Discounts 🧮

Enter the final price in dollars (just the number).

1) A $25 shirt is marked up $20\%$ . New price $= \$ ,? $**2)** An \$ 80 jacket is $30\%$ off. Sale price $= \$ ,? $**3)** A \$ 200 bike is $40\%$ off. Sale price $= \$ ,?$

Working Backward (Finding the Original)

If you know the final price and the percent change, divide to recover the original.

Worked Example: A jacket costs $48 after $20\%$ off. What was the original price?

You paid $80\%$ of the original, so:

$0.80 \times (\text{original}) = 48 \quad\Rightarrow\quad \text{original} = \frac{48}{0.80} = \$60$

⚠️ Don't just add $20\%$ back to $48. That gives $57.60, which is wrong, because the $20\%$ was taken off the larger original price ( $60$ ), not the sale price ( $48$ ). Always divide by the multiplier.

Worked Example: $8\%$ sales tax on a $45 purchase

$\text{tax} = 0.08 \times 45 = \$3.60$ $\text{total} = 45 + 3.60 = \$48.60$

Worked Example: An $18\%$ tip on a $50 meal

$\text{tip} = 0.18 \times 50 = \$9.00$ $\text{total} = 50 + 9 = \$59.00$

💡 Shortcut: since tax and tip are added on, you can multiply the base by $(1 + \text{rate})$ for the total in one step: $1.08 \times 45 = \$ 48.60$.

Concept Check 🎯

Commission

A commission is a percent of sales that a salesperson earns as pay.

$\text{commission} = \text{rate} \times \text{total sales}$

Worked Example: A realtor earns $6\%$ commission on a $250{,}000 home

$0.06 \times 250{,}000 = \$15{,}000$

Worked Example: A clerk earns $5\%$ commission on $1{,}200 in sales

$0.05 \times 1{,}200 = \$60$

💡 Commission is the same operation as tax and tip — "percent of a number" — it's just money earned instead of money added to a bill.

Tax, Tip, and Commission 🧮

Enter dollar amounts (just the number).

1) A $7.5\%$ tax on a $120 item. Tax $= \$ ,? $**2)** A$ 15% $tip on a \$ 40 meal. Tip $= \$ ,? $**3)** A$ 6% $commission on \$ 1{,}200 in sales. Commission $= \$ ,?$

Simple Interest

Simple interest is money earned (or owed) on a starting amount, computed with one formula:

$I = P \cdot r \cdot t$

Symbol	Meaning
$I$	interest earned (in dollars)
$P$	principal — the starting amount
$r$	annual rate, as a decimal
$t$	time, in years

Worked Example: $1{,}000 at $5\%$ for $2$ years

$I = 1000 \times 0.05 \times 2 = \$100$

The total amount in the account is principal plus interest:

$P + I = 1000 + 100 = \$1{,}100$

⚠️ Two common slips: (1) use $r$ as a decimal ( $5\% = 0.05$ , not $5$ ), and (2) keep $t$ in years (6 months $= 0.5$ year).

Simple Interest 🔽

Use $I = P \cdot r \cdot t$ for $500 at $4\%$ for $3$ years.

⚠️ Remember the big three traps: convert percents to decimals before multiplying, divide by the original for percent change, and use $r$ as a decimal with $t$ in years.

Mixed Review 🔽

From Recognizing to Computing

The review above asked you to recognize the right form. Now let's compute full answers. Keep the playbook handy:

"What is X% of N?" → multiply.
"P is X% of what?" → divide $P$ by the decimal.
A tip total → multiply by $1 + \text{rate}$ .

🔑 Decimal-ize the percent first, every time. Then the arithmetic is easy.

Mixed Practice 🧮

1) What is $35\%$ of $200$ ? $\,?$ 2) $90$ is $75\%$ of what number? $\,?$ 3) A $45 meal with a $20\%$ tip costs a total of $\$ ,?$

Application Challenge 🎯

Exit Quiz ✅

Answer all three to finish the lesson.

\;0.70 \times 80 = \

Percents and Applications

Percents and Applications - Complete Interactive Lesson

Part 1: What a Percent Really Means

💯 Percents and Applications

Topics in This Part

Percent = "Per Hundred"

Percents ↔ Decimals

Percents ↔ Fractions

Part 2: Finding a Percent of a Number

💯 Percents and Applications

Part 3: Percent Increase & Decrease

💯 Percents and Applications

Part 4: Real-World Applications

💯 Percents and Applications

Sales Tax and Tips

Part 5: Mixed Practice & Mastery Check

💯 Percents and Applications

Quick Reference

Memorize These Benchmarks

"Of" Means Multiply

Worked Example: 15%15\%15% of 606060

Worked Example: 8%8\%8% of 250250250

Worked Example: 150%150\%150% of 404040

Estimate to Catch Mistakes

The Three Percent Questions

Worked Example: "What percent of 242424 is 181818?"

Worked Example: "303030 is 25%25\%25% of what number?"

Decoding the Words

Finding the Percent Change

Worked Example: A price rises from 404040 to 505050

Worked Example: A price drops from 808080 to 606060

Markups and Discounts

Method A — Find the change, then add or subtract

Method B — Multiply by the final percent (the shortcut)

One Multiplier, Final Answer

Working Backward (Finding the Original)

Worked Example: A jacket costs $48 after 20%20\%20% off. What was the original price?

Worked Example: 8%8\%8% sales tax on a $45 purchase

Worked Example: An 18%18\%18% tip on a $50 meal

Commission

Worked Example: A realtor earns 6%6\%6% commission on a $250{,}000 home

Worked Example: A clerk earns 5%5\%5% commission on $1{,}200 in sales

Simple Interest

Worked Example: $1{,}000 at 5%5\%5% for 222 years

From Recognizing to Computing

Worked Example: $15\%$ of $60$

Worked Example: $8\%$ of $250$

Worked Example: $150\%$ of $40$

Worked Example: "What percent of $24$ is $18$ ?"

Worked Example: " $30$ is $25\%$ of what number?"

Worked Example: A price rises from $40$ to $50$

Worked Example: A price drops from $80$ to $60$

Worked Example: A jacket costs $48 after $20\%$ off. What was the original price?

Worked Example: $8\%$ sales tax on a $45 purchase

Worked Example: An $18\%$ tip on a $50 meal

Worked Example: A realtor earns $6\%$ commission on a $250{,}000 home

Worked Example: A clerk earns $5\%$ commission on $1{,}200 in sales

Worked Example: $1{,}000 at $5\%$ for $2$ years